Properties

Label 6229.a.6229.1
Conductor 6229
Discriminant 6229
Mordell-Weil group \(\Z \times \Z\)
Sato-Tate group $\mathrm{USp}(4)$
\(\End(J_{\overline{\Q}}) \otimes \R\) \(\R\)
\(\End(J_{\overline{\Q}}) \otimes \Q\) \(\Q\)
\(\overline{\Q}\)-simple yes
\(\mathrm{GL}_2\)-type no

Related objects

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Minimal equation

Minimal equation

Simplified equation

$y^2 + (x^3 + x + 1)y = -2x^4 + 3x^3 - 3x^2$ (homogenize, simplify)
$y^2 + (x^3 + xz^2 + z^3)y = -2x^4z^2 + 3x^3z^3 - 3x^2z^4$ (dehomogenize, simplify)
$y^2 = x^6 - 6x^4 + 14x^3 - 11x^2 + 2x + 1$ (minimize, homogenize)

magma: R<x> := PolynomialRing(Rationals()); C := HyperellipticCurve(R![0, 0, -3, 3, -2], R![1, 1, 0, 1]);
 
sage: R.<x> = PolynomialRing(QQ); C = HyperellipticCurve(R([0, 0, -3, 3, -2]), R([1, 1, 0, 1]));
 
magma: X,pi:= SimplifiedModel(C);
 
sage: X = HyperellipticCurve(R([1, 2, -11, 14, -6, 0, 1]))
 

Invariants

Conductor: \( N \)  =  \(6229\) = \( 6229 \)
magma: Conductor(LSeries(C)); Factorization($1);
 
Discriminant: \( \Delta \)  =  \(6229\) = \( 6229 \)
magma: Discriminant(C); Factorization(Integers()!$1);
 

Igusa-Clebsch invariants

Igusa invariants

G2 invariants

\( I_2 \)  = \(-120\) =  \( - 2^{3} \cdot 3 \cdot 5 \)
\( I_4 \)  = \(11172\) =  \( 2^{2} \cdot 3 \cdot 7^{2} \cdot 19 \)
\( I_6 \)  = \(-507864\) =  \( - 2^{3} \cdot 3 \cdot 7 \cdot 3023 \)
\( I_{10} \)  = \(25513984\) =  \( 2^{12} \cdot 6229 \)
\( J_2 \)  = \(-15\) =  \( - 3 \cdot 5 \)
\( J_4 \)  = \(-107\) =  \( - 107 \)
\( J_6 \)  = \(389\) =  \( 389 \)
\( J_8 \)  = \(-4321\) =  \( - 29 \cdot 149 \)
\( J_{10} \)  = \(6229\) =  \( 6229 \)
\( g_1 \)  = \(-759375/6229\)
\( g_2 \)  = \(361125/6229\)
\( g_3 \)  = \(87525/6229\)

magma: IgusaClebschInvariants(C); IgusaInvariants(C); G2Invariants(C);
 
sage: C.igusa_clebsch_invariants(); [factor(a) for a in _]
 

Automorphism group

\(\mathrm{Aut}(X)\)\(\simeq\) $C_2$
magma: AutomorphismGroup(C); IdentifyGroup($1);
 
\(\mathrm{Aut}(X_{\overline{\Q}})\)\(\simeq\) $C_2$
magma: AutomorphismGroup(ChangeRing(C,AlgebraicClosure(Rationals()))); IdentifyGroup($1);
 

Rational points

Known points
\((1 : 0 : 0)\) \((1 : -1 : 0)\) \((0 : 0 : 1)\) \((0 : -1 : 1)\) \((1 : -1 : 1)\) \((1 : -2 : 1)\)
\((3 : -4 : 1)\) \((2 : -17 : 3)\) \((5 : -25 : 2)\) \((3 : -27 : 1)\) \((2 : -36 : 3)\) \((5 : -128 : 2)\)

magma: [C![0,-1,1],C![0,0,1],C![1,-2,1],C![1,-1,0],C![1,-1,1],C![1,0,0],C![2,-36,3],C![2,-17,3],C![3,-27,1],C![3,-4,1],C![5,-128,2],C![5,-25,2]];
 

Number of rational Weierstrass points: \(0\)

magma: #Roots(HyperellipticPolynomials(SimplifiedModel(C)));
 

This curve is locally solvable everywhere.

magma: f,h:=HyperellipticPolynomials(C); g:=4*f+h^2; HasPointsEverywhereLocally(g,2) and (#Roots(ChangeRing(g,RealField())) gt 0 or LeadingCoefficient(g) gt 0);
 

Mordell-Weil group of the Jacobian

Group structure: \(\Z \times \Z\)

magma: MordellWeilGroupGenus2(Jacobian(C));
 

Generator $D_0$ Height Order
\(2 \cdot(1 : -1 : 1) - (1 : -1 : 0) - (1 : 0 : 0)\) \((x - z)^2\) \(=\) \(0,\) \(y\) \(=\) \(-xz^2\) \(0.303214\) \(\infty\)
\((1 : -1 : 1) - (1 : -1 : 0)\) \(z (x - z)\) \(=\) \(0,\) \(y\) \(=\) \(-z^3\) \(0.053190\) \(\infty\)

2-torsion field: 6.2.398656.1

BSD invariants

Hasse-Weil conjecture: unverified
Analytic rank: \(2\)
Mordell-Weil rank: \(2\)
2-Selmer rank:\(2\)
Regulator: \( 0.015602 \)
Real period: \( 18.54212 \)
Tamagawa product: \( 1 \)
Torsion order:\( 1 \)
Leading coefficient: \( 0.289307 \)
Analytic order of Ш: \( 1 \)   (rounded)
Order of Ш:square

Local invariants

Prime ord(\(N\)) ord(\(\Delta\)) Tamagawa L-factor
\(6229\) \(1\) \(1\) \(1\) \(( 1 + T )( 1 - 125 T + 6229 T^{2} )\)

Sato-Tate group

\(\mathrm{ST}\)\(\simeq\) $\mathrm{USp}(4)$
\(\mathrm{ST}^0\)\(\simeq\) \(\mathrm{USp}(4)\)

Decomposition of the Jacobian

Simple over \(\overline{\Q}\)

Endomorphisms of the Jacobian

Not of \(\GL_2\)-type over \(\Q\)

Endomorphism ring over \(\Q\):

\(\End (J_{})\)\(\simeq\)\(\Z\)
\(\End (J_{}) \otimes \Q \)\(\simeq\)\(\Q\)
\(\End (J_{}) \otimes \R\)\(\simeq\) \(\R\)

All \(\overline{\Q}\)-endomorphisms of the Jacobian are defined over \(\Q\).